Measurement 92 (2016) 382–390

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Measurement

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On-line three-dimensional point cloud data extraction method forscan-tracking measurement of irregular surface using bi-Akima spline

http://dx.doi.org/10.1016/j.measurement.2016.06.0080263-2241/� 2016 Elsevier Ltd. All rights reserved.

⇑ Corresponding author at: School of Manufacturing Science & Engineering,Sichuan University, No. 24 South Section 1, Yihuan Road, Chengdu 610065, China.

E-mail address: yetao@scu.edu.cn (Y. Tao).

Ye Tao a,⇑, Yong-Qing Wang b, Hai-Bo Liu b, Meng Li c

a School of Manufacturing Science and Engineering, Sichuan University, Chinab School of Mechanical Engineering, Dalian University of Technology, Chinac FAW-Volkswagen Automotive Company, Ltd, China

a r t i c l e i n f o

Article history:Received 20 July 2013Received in revised form 30 May 2016Accepted 7 June 2016Available online 22 June 2016

Keywords:Point cloud dataData extractionScan-tracking measurementBi-Akima spline interpolation

a b s t r a c t

Point cloud data extraction is an important process in scan-tracking measurement. In this paper, a newmethod of on-line three-dimensional point cloud data extraction for scan-tracking measurement is pro-posed for reducing extremely dense sampled data while maintaining data accuracy during the real-timescan-tracking measuring process. It is inspired from sketch paintings: First outlining the broad contour ofthe curve and then revising local details till the interpolated curve satisfies the required accuracy. Thismethod adopts bi-Akima spline interpolation for connecting acquired points in NC machining or for pointdata fitting in reverse engineering. It can reduce efficiently the amount of point data with a smaller datareduction ratio and a smoother machined/fitted surface than conventional three-dimensional chordalmethod.

� 2016 Elsevier Ltd. All rights reserved.

1. Introduction

Surface scan-tracking measurement technology is one of thekey technologies in Numerical Control (NC) copying manufacturesystems and is also an important method for processing irregularsurface parts in reverse engineering [1–5]. Experts and researchersall over the world are concerned about this technology increas-ingly. Point cloud data acquisition is an important step in scan-tracking measuring process [4,5]. Nowadays, methods for acquir-ing point cloud data used in scan-tracking measurement mainlyinclude contact measurement and non-contact measurement.There are problems in using these acquiring methods since theyproduce extremely dense point data at a great rate, and not all ofthese point data is necessary [6]. Moreover, bottlenecks are createdowing to the inefficiencies in storing and manipulating them.Describing measuring objects with the least point cloud informa-tion is in expectation. In general, the larger curvature change is,the denser the point cloud is, and vice versa. Thus, a high-efficiency, high-quality point cloud data reduction method is beingpursued all the time [7]. Sampling algorithms for scan-trackingmeasurement generally include isochronous sampling and equidis-tant sampling. These algorithms are easy to implement but cannotadjust the number of sampled points according to the changes of

surface curvature, which consequently leads to the loss of someimportant geometric information. Consequently, they cannot meetthe requirements of guaranteeing precision and saving memory atthe same time, and only suitable for the condition of the curvaturechanging little or nothing [5].

In response to the above situation, point cloud data reductionmethods are currently being studied by many researchers all overthe world. Lee et al. [7] introduced a procedure for handling pointcloud data acquired by laser scanners. This method uses one-directional or bi-directional non-uniform grid to reduce the datasize. Fujimoto and Kariya [8] proposed an improved sequentialmethod using an angle parameter for data reduction which cancontrol the distance between input and output data. This methodalso it possible to accept a large amount of data in a small-size sys-tem. Chen et al. [9] suggested a method to reduce the point data byreducing the number of triangles required in a polyhedral model.They demonstrated their algorithm by reducing the number of tri-angles in an STL file of a human face digitised by a CMM machine.Martin et al. [10] proposed a data reduction method by using a uni-form grid in their EU Copernicus project. Their method uses a ‘‘me-dian filtering” approach, which has been widely used in imageprocessing. Hamann [11] presented a method of data reductionfor triangulation files based on an iterative triangle removal princi-ple. As a measure of file size reduction, each triangulation isweighted on the basis of the principal curvature estimates at itsvertices and interior angles. Hamann and Chen [12] proposed a

Y. Tao et al. /Measurement 92 (2016) 382–390 383

method to reduce the point data in making various planar curves,compressing 2D images, and visualising volumes.vMajor researchefforts of above methods focus on manipulating polyhedral mod-els. Various schemes are also used to reduce the amount of pointdata from the initial point clouds. However, none of these methodscan achieve on-line point data reduction during the real-time mea-suring process, they could only reduce the overall point cloud datafor post-processing after data acquisition process.

Up to date, only chordal method can achieve on-line point clouddata extraction of the cross-sectional curve during the real-timedata acquisition process [5,6]. It has the ability of accepting andrejecting sampled data reasonably based on the curvature changesand it can guarantee a required accuracy with less sampled data.Unfortunately, this method assumes that adjacent acquired pointsare connected by a straight line, which specifies the interpolationform must be a linear type in machining process and in data pro-cessing process. Conversely, if we use spline curves (e.g. NURBSspline) to connect these points acquired by chordal method, therequired accuracy cannot be guaranteed. Moreover, when usinglinear interpolation to connect discrete dense sets of sampledpoints in NC machining, the curve is not smooth and there aremany cusp points, thus leads to sudden changes in feed velocityand acceleration of NC servo systems [13]. Machining accuracyand surface quality of mating surfaces will be affected to someextent. In addition, the reduction ratio of the chordal method needsto be further improved.

To avoid these problems, this paper presents an on-line three-dimensional point cloud data extraction method for scan-tracking measurement of irregular surface using bi-Akima splineinterpolation, which can reduce the amount of point data acquiredduring the real-time data sampling process.

2. A brief description of chordal method

2.1. Two-dimensional chordal method [5,6]

For ease of understanding and for convenience, we first describetwo-dimensional chordal method. The basic principle of thismethod is shown in Fig. 1. At first, the scan-tracking measuringsystem records the coordinate value of the first point Pi as the ini-tial reference point. Then in every sampling period, the control sys-tem samples the current coordinate value and works out themaximum chord height hmax from the chord that connects currentsampling point with previous recorded reference point to the arctrajectory. The maximum chord height hmax is made by computingthe corresponding chord heights hi+1, hi+2, . . ., hi+m, . . ., hi+n�1 of allthe points between the current sampling point Pi+n and the refer-ence point Pi. The chord height hi+m at point Pi+m can be successfullyevaluated as Eq. (1):

hiþm ¼ Aðxiþm � xiÞ � Bðyiþm � yiÞ�� ��� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2 þ B2q

; ð1Þ

Fig. 1. Two-dimensional chordal method for cross-sectional curve.

where A = yi+n � yi, B = xi+n � xi, (xi, yi) is the coordinate value ofpoint Pi, (xi+m, yi+m) is the coordinate value of point Pi+m, (xi+n, yi+n)is the coordinate value of point Pi+n. Then taking the maximum ofall the chord heights hmax to compare with the required accuracye. If hmax is less than the required accuracy e, the correspondingsampling point will not be recorded. While the point Pi+n�1 sampledin the former sampling period should be recorded in the point clouddata file. After that, this recorded point will be used as the new ref-erence point to repeat the above cycle until the termination of thesampling process.

2.2. Three-dimensional chordal method

Two-dimensional chordal method can only reduce the pointdata of certain cross-sectional curve in two-dimensional plane,but cannot handle the point cloud data of complex surface inthree-dimensional space. Therefore, three-dimensional chordalmethod is proposed in this paper. It is an extension of two-dimensional chordal method. As shown in Fig. 2, the only differ-ence between these two methods is that the chord height hi+m atpoint Pi+m for three-dimensional chordal method is calculated byEq. (2),

hiþm ¼ PiPiþn � PiPiþm

�� ��� PiPiþn

�� ��; ð2Þwhere the coordinate value of point Pi, Pi+m, Pi+n is (xi, yi, zi),(xi+m, yi+m, zi+m), (xi+n, yi+n, zi+n), respectively.

The chordal method achieves data reduction under the premiseof ensuring accuracy (i.e. the required accuracy e). But it can onlyuse a polyline connected by the recorded points to approximatethe original curve. Therefore, for ensuring the machining accuracyof a mating surface, the NC interpolation form must be a lineartype rather than spline curve type. When linear interpolation isused to connect dense point cloud data, unexpected oscillation ofthe mechanical system may appear in NC real-time interpolationprocess, which may affect the machining accuracy and surfacequality of mating surfaces.

If we choose a spline curve fitting method to connect therecorded points and to accept and reject the sampled data accord-ing to the maximum deviation between the fitting curve with theoriginal sampling point, there will be no edges and corners in thesmooth cutter path of NC machining process, which can effectivelyavoid the oscillation of the mechanical system. What is more, thereduction ratio of point cloud data might be further reduced.

3. Overview of spline interpolation for NC machining

There are currently three types of spline interpolation sup-ported by commercial NC systems [13], as shown in Fig. 2.

(i) NURBS (non-uniform, rational basis) spline. The NURBSspline does not pass directly through the data points [14].The programmed positions are merely a set of control points

Fig. 2. Comparison of three spline types with identical interpolation points.

384 Y. Tao et al. /Measurement 92 (2016) 382–390

of the spline. NURBS spline is the optimum means for defin-ing tool paths on sculptured surfaces. The primary purpose isto act as the interface to CAD systems. A third degree NURBSspline does not produce any oscillations in spite of its con-tinuously curved transitions.

(ii) Akima spline. The Akima spline passes exactly through allthe data points, and it produces virtually no unnatural wig-gles [15]. A polynomial of third degree is used for interpola-tion. The Akima spline is local, a change to an interpolationpoint affects only up to six adjacent points. The primaryapplication for this spline type is therefore the interpolationof discrete digitized points.

(iii) Cubic spline. In contrast to Akima spine, the Cubic splinealso passes exactly through all the data points, and it usesthird degree polynomials. But it is not local, changes to aninterpolation point can influence a large number of blocks(with gradually decreasing effect). It tends to have unex-pected fluctuations. However, it can be used in cases wherethe interpolation points lie along an analytically calculatedcurve.

4. On-line point cloud data extraction using bi-Akima splineinterpolation

By comparison of different spline interpolation types for NCmachining in Section 3, we choose Akima spline to connectrecorded points obtained by scan-tracking measurement since itis the best choice for the interpolation of discrete digitized points.And for reducing the point cloud data of complex surface in three-dimensional space, finally, bi-Akima spline interpolation isadopted in this paper.

4.1. brief description of bi-Akima spline interpolation

The bi-Akima method is based on a piecewise function com-posed of a set of polynomials, each of degree three. The coefficientsof each polynomial between a pair of given points are determinedby the coordinates of and the slopes at these two points. The slopeof the curve at each given point is determined locally by the coor-dinates of five points, with the point in question as a center point,and two points on each side of it. Akima assumed that the slope ty,i,tz,i of the curve at point Pi(Xi, Yi, Zi) is determined by

ty;i ¼ jmy;iþ1�my;i jmy;i�1þjmy;i�1�my;i�2 jmy;i

jmy;iþ1�my;i j�jmy;i�1�my;i�2 j

tz;i ¼ jmz;iþ1�mz;i jmz;i�1þjmz;i�1�mz;i�2 jmz;ijmz;iþ1�mz;i j�jmz;i�1�mz;i�2 j

8<: ; ð3Þ

where my,i and mz,i are the slopes of projections of line segmentPiPiþ1 in XOY and XOZ plane, respectively, i = 1, 2, 3, . . ., n. The slopety,i and tz,i depend only on the slopes of four line segmentsPi�2Pi�1; Pi�1Pi; PiPiþ1 and Piþ1Piþ2� �

and are independent of theinterval widths. At each end and beginning of the curve, two morepoints have to be estimated from the given points. According to thetrend of the given points, Akima put forward this assumption:

my;0 ¼ 2my;1 �my;2

my;�1 ¼ 2my;0 �my;1

my;n ¼ 2my;n�1 �my;n�2

my;nþ1 ¼ 2my;n �my;n�1

8>>><>>>:

and

mz;0 ¼ 2mz;1 �mz;2

mz;�1 ¼ 2mz;0 �mz;1

mz;n ¼ 2mz;n�1 �mz;n�2

mz;nþ1 ¼ 2mz;n �mz;n�1

8>>><>>>:

: ð4Þ

The polynomial can be uniquely determined according to themethod above, and each piecewise interpolation function can bewritten in the following form:

y ¼ Ay þ Byðx� XiÞ þ Cyðx� XiÞ2 þ Dyðx� XiÞ3

z ¼ Az þ Bzðx� XiÞ þ Czðx� XiÞ2 þ Dzðx� XiÞ3

(; ð5Þ

in which

AyðiÞ ¼ Yi

ByðiÞ ¼ ty;i

CyðiÞ ¼ 3my;i�2ty;i�ty;iþ1Xiþ1�Xi

DyðiÞ ¼ ty;iþty;iþ1�2my;i

ðXiþ1�XiÞ2

8>>>>><>>>>>:

and

AzðiÞ ¼ Zi

BzðiÞ ¼ tz;i

CzðiÞ ¼ 3mz;i�2tz;i�tz;iþ1Xiþ1�Xi

DzðiÞ ¼ tz;iþtz;iþ1�2mz;i

ðXiþ1�XiÞ2

8>>>>><>>>>>:

ð6Þ

Thus a complete curve is obtained, which is composed of sev-eral piecewise polynomials and passes through every chosen pointwithout unnatural wiggles.

4.2. The principle of point cloud data extraction

This data extraction method is inspired from sketch paintings.First, the broad contours of the curve are determined, and thenlocal details are revised gradually, until the interpolated curve sat-isfies the required accuracy. The overall point cloud data extractionflowchart is shown in Fig. 3.

For purpose of depicting the broad contours of a curve using acertain sequence of points, first, initial feature points should bechosen. As shown in Fig. 4, we require that the initial feature pointsinclude the following types of points: (i) Start points and endpoints. (ii) The inflection points of the curve. At these points, thesecond derivative of the curve is equal to zero and the third deriva-tive is not. These points are the divides of the curve concave andconvex portions. (iii) The cusp points of the curve. At these points,the third derivative values are equal to zero. These kinds of pointsare selected rather than the extreme points, since these points arelocated in the position where the curve has the local extrema of thecurvature. We define it as the cusp of the curve. If we connect twoadjacent inflection points to get a line segment, the cusp betweenthese two inflection points will be the farthest points to the linesegment. In some curves, sine curve for instance, the cusp andthe extreme points are in the same location, but in most irregularsampled curves, they are not. In this condition, the distance fromthe cusp point to the line segment is the largest. Hence, they can

Fig. 3. Flowchart of point cloud data extraction.

Fig. 4. The selection of initial feature points.

Y. Tao et al. /Measurement 92 (2016) 382–390 385

reflect the maximum extent of the concave and convex. So far, thebroad contour of any curve can be roughly described by the initialfeature points of the above three types.

The original set of sampled points Qj (j = 1, 2, 3, . . .,m) is not acontinuous curve, so the derivative function cannot directly be cal-culated. But the point cloud data is dense enough in scan-trackingmeasurement, so that we can approximate the derivative as

y0j ¼yjþ1�yjxjþ1�xj

y00j ¼y0jþ1

�y0j

xjþ1�xj

y000j ¼ y00jþ1

�y00j

xjþ1�xj

8>>>><>>>>:

and

z0j ¼zjþ1�zjxjþ1�xj

z00j ¼z0jþ1

�z0j

xjþ1�xj

z000j ¼ z00jþ1

�z00j

xjþ1�xj

8>>>><>>>>:

; ð7Þ

where y0j; y00j ; y000j are the derivatives of projection curve in XOY planeand z0j; z00j ; z000j are the derivatives of projection curve in XOZ plane.

Note that the original discrete points are obtained from physicalworkpieces, when calculate inflection points and cusp points, it isdifficult to obtain points where the derivative is exactly equal tozero (e.g. y00j ¼ 0; z00j ¼ 0; y000j ¼ 0 or z000j ¼ 0). Therefore, we stipulatethat the zero-crossing point is the zero points. Take the secondderivative of projection curve in XOY plane as an example: Ify00j � y00jþ1 < 0, we consider point Qj as the inflection point.

Upon completion of initial feature points selection, the initialfeature points can be used as the selected points Pi(Xi, Yi, Zi) forthe bi-Akima interpolation. These selected points divide the curveinto several intervals, and each interval is described as a cubicpolynomial. As shown in Fig. 5, the interpolation curve passesthrough the selected points Pi(Xi, Yi, Zi) and Pi+1(Xi+1, Yi+1, Zi+1) withno deviation. Apart from the selected points, there is deviation hjbetween each original sampled point Qj and the interpolationcurve. Herein, the deviation hj means the shortest distance fromevery original sampled point Qj to the interpolation curve.

Fig. 5. Schematic diagram of point

The distance s can be expressed as

s ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xjÞ2 þ ðy� yjÞ2 þ ðz� zjÞ2

q; ð8Þ

where point Qj(xj, yj, zj) can be any original sampled points betweenPi(Xi, Yi, Zi) and Pi+1(Xi+1, Yi+1, Zi+1), and point Pcurv(x, y, z) is the pointin the interpolation curve that makes the distance s shortest. Thusthe deviation hj of every original sampled point Qj can be calculatedfrom Eq. (9):

hj ¼ MINx2ðXi ;Xiþ1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xjÞ2 þ ðy� yjÞ2 þ ðz� zjÞ2

q� �: ð9Þ

The next step is to seek the maximum deviation hmax. In mostcases, the maximum deviation of hj is near the middle positionbetween two selected points Pi and Pi+1, but it is difficult to findthe exact location directly. The general method is to calculate thedeviation hj of all the points thereby obtaining the maximum devi-ation, or just consider the deviation of middle point as the maxi-mum deviation [5]. The former is too cumbersome and the latteris not accurate. In order to simplify the process, we proposed themethod below. First, take the middle point Qj and two adjacentpoints Qj�1 and Qj+1 to calculate the deviation hj, hj�1 and hj+1 asshown in Fig. 5. If hj is the maximum of three, then we identifiedhj is the maximum deviation hmax in this interval. Otherwise, com-pare hj�1 with hj+1 (e.g. hj+1 > hj�1), then along the direction of a lar-ger value (hj+1), calculate and compare the deviation at eachoriginal point (hj+1, hj+2, hj+3, . . .) until the deviation become smal-ler. The point before the smaller one is the max deviation pointin this interval, and we also get the max deviation value (e.g.,hj+4 < hj+3, hj+3 = hmax). After that, compare the max deviation valueof all intervals and obtain the max deviation of the whole curve.The max deviation h is then compared to the required accuracy e.If h > e, the point where gets h will be added in the selected points.Then repeat the process above till h 6 e. Finally, we get the com-pressed point set.

5. Experimental results

In this section, the proposed point cloud data extraction methodwas tested in the real-time scan-tracking measuring process. Themeasuring system is composed of a vertical lathes (Fig. 6) and acommercial CNC system of SINUMERIK 840D.

Fig. 7 shows the overall architecture of the measuring system. Itexplains how the proposed algorithm is embedded in a commercialopen CNC system with scan-tracking measuring function.

The Sinumerik 840D, a commercial open architecture CNC sys-tem, was used as a prototype system for the embedding test. Itconsists of the NCU which takes charge of motion control,

cloud data extraction method.

Fig. 6. Scan-tracking measuring system.

Fig. 7. Architecture of the measuring system.

386 Y. Tao et al. /Measurement 92 (2016) 382–390

Y. Tao et al. /Measurement 92 (2016) 382–390 387

interpolation and all other real-time tasks, the PLC based on Step7to deal with logic controls, and the MMC which simplifies themanipulation of the machine for operators. MMC supports duplexserial communication with the NCU through the MPI bus.

OEM applications was developed under WINDOWS with theOEM package and downloaded to the MMC. Our proposed method(bi-Akima method) and chardal method were both embedded intothe OEM application. Either of the two methods can be chosen dur-ing the data acquisition process.

As shown in Fig. 8, the measured workpiece is a half-ellipsoidalsurface which is welded together by seven pieces of thin-walledaluminum alloy sheet. It is a typical irregular surface with geomet-ric abrupt transition changes in the welding line area. The semi-major axis and the semi-minor axis of the ellipse are 1450 mmand 950 mm respectively. The width and height of the welding lineare about 10 mm and 3 mm, respectively. Rotational progressivescanning mode was selected, the scan-tracking movement was car-ried out with period feed along the ellipsoidal surface, and the dis-tance between adjacent scanning lines is 7 mm. The scanningprobe DIGIT-02 based on LVDT was adopted in this measuring sys-tem. The measuring range of the probe is ±1 mm, and the linearregion of measuring range is ±0.8 mm. A nonlinear cubic polyno-mial calibration was carried out before measurement. The error

Fig. 8. Scan-tracking measurement of irregular surface.

Fig. 9. Initial point cloud data acquisition

in each axis of the scanning measurement system can be controlledat less than 0.016 mm.

Fig. 9 shows the initial point cloud data acquisition results byisochronous sampling method, and there are a total of 272,638acquired points for measuring the whole surface.

After the initial point cloud data was acquired, data reductioncomparison is made between three-dimensional chordal methodand bi-Akima extraction method under different requiredaccuracy.

Table 1 summarizes the results of reduction performance andgives the comparison between the output of bi-Akima methodand chordal method. In order to make the comparison more intu-itive, Fig. 10 shows the difference of reduced point sets betweenthese two methods by displaying the spatial distribution. Fig. 11gives the comparison of reduction ratio between these two meth-ods under different required accuracy (i.e. from 0.001 mm to1 mm). As the required accuracy decreases, the reduction ratioincreases for both methods; however, for all levels of requiredaccuracy, the bi-Akima reduction method shows lower reductionratio and more uniform spatial distribution than chordal method.Furthermore, the reduction performance of bi-Akima extractionmethod is superior to chordal method especially in the case of highaccuracy requirement (e.g. required accuracy e = 0.001 mm, thedata reduction ratio of the proposed method and chordal methodare 44.27% and 87.06%, respectively).

using isochronous sampling method.

Table 1Comparison of reduction performance under different required accuracy.

Requiredaccuracy/mm

Number of points Reduction ratio/%

Chordalmethod

Bi-Akimamethod

Chordalmethod

Bi-Akimamethod

0.001 237,363 120,702 87.06 44.270.002 189,824 118,740 69.62 43.550.005 152,674 108,389 55.99 39.760.01 136,027 91,596 49.89 33.600.02 123,891 69,594 45.44 25.530.05 103,205 42,198 37.85 15.480.1 87,008 26,681 31.91 9.790.2 61,124 12,061 22.42 4.420.5 28,473 5309 10.44 1.941 9029 3766 3.31 1.38

Fig. 10. Spatial distributions of point cloud extracted by different methods under different required accuracy e: (a) chordal method, e = 0.001 mm, (b) bi-Akima method,e = 0.001 mm, (c) chordal method, e = 0.01 mm, (d) bi-Akima method, e = 0.01 mm, (e) chordal method, e = 0.1 mm, (f) bi-Akima method, e = 0.1 mm, (g) chordal method,e = 1 mm, and (h) bi-Akima method, e = 1 mm.

Fig. 11. Data reduction ratio under different required accuracy.

388 Y. Tao et al. /Measurement 92 (2016) 382–390

Fig. 12 shows the deviations between original sampled pointsand interpolation curve under different required accuracies. Bothmethods can strictly control the deviation within the error toler-ance range (i.e., the deviation between each original sampled pointand interpolation curve is less than or equal to required accuracy).

However, the deviation of the proposed method is more evenlydistributed then the chordal method under the same requiredaccuracy.

In this comparative test, using the same original sampled data,the proposed point cloud data extraction method shows better per-formance in data reduction ratio and deviation distribution thantraditional three-dimensional chordal method. The comparisonresults confirm the effectiveness of the new method proposed inthis paper. And it is worth mentioning that the reduction ratio offreeform surface point cloud data depends not only on the reduc-tion method, but also on the required accuracy and the numberof original sampled points.

Experimental results indicated that the proposed bi-Akimareduction method was able to obtain a smaller data reduction ratioand a smoother machined surface than conventional methods. Thereduction performance of this method was superior to chordalmethod obviously under the same required accuracy. It can beused in the data acquisition process of scan-tracking measurementto replace traditional on-line point data extraction methods.

Fig. 12. Comparison of deviations under different required accuracy e: (a) chordal method, e = 1 mm, (b) bi-Akima method, e = 1 mm, (c) chordal method, e = 0.1 mm, (d) bi-Akima method, e = 0.1 mm, (e) chordal method, e = 0.01 mm, and (f) bi-Akima method, e = 0.01 mm.

Y. Tao et al. /Measurement 92 (2016) 382–390 389

6. Conclusion

In this paper, an on-line three-dimensional point cloud dataextraction method for scan-tracking measurement of irregular sur-face was proposed and implemented. This method could reduceextremely dense sampled data while maintaining data accuracyduring the real-time scan-tracking measuring process. Experimen-tal results indicated that the proposed bi-Akima reduction methodwas able to obtain a smaller data reduction ratio and a smoothermachined surface than conventional methods. The reduction per-formance of this method was superior to chordal method obviously

under the same required accuracy. It can be used in the data acqui-sition process of scan-tracking measurement to replace traditionalon-line point data extraction methods.

Acknowledgments

This project is supported by the National Natural ScienceFoundation of China (Program Grant No. 51505310), the NationalNatural Science Foundation of China (Program Grant No.51435011) and Advanced Research Foundation (Program GrantNo. 9140A18020310JW0902).

390 Y. Tao et al. /Measurement 92 (2016) 382–390

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